Quadrature imbalance estimation using unbiased training sequences

ABSTRACT

A system and method are provided for removing quadrature imbalance errors in received data. The method accepts an unbiased training sequence in a quadrature demodulation receiver. An unbiased training sequence has a uniform accumulated power evenly distributed in a complex plane, and includes predetermined reference signals (p) at frequency +f and predetermined mirror signals (p m ) at frequency −f. The unbiased training sequence is processed, generating a sequence of processed symbols (y) at frequency +f, representing complex plane information in the unbiased training sequence. Each processed symbol (y) is multiplied by the mirror signal (p m ), and an unbiased quadrature imbalance estimate B m  is obtained at frequency (−f). Using quadrature imbalance estimates, channel estimates, and processed symbols, an imbalance-corrected symbol can be generated.

CLAIM OF PRIORITY UNDER 35 U.S.C. §119

The present application for patent claims priority to ProvisionalApplication No. 60/896,480, filed Mar. 22, 2007, entitled, QUADRATUREIMBALANCE MITIGATION USING UNBIASED TRAINING SIGNALS, status Pending.

CLAIM OF PRIORITY UNDER 35 U.S.C. §120

The present application for patent is a continuation of patentapplication Ser. No. 11/853,808, filed Sep. 11, 2007, entitled,QUADRATURE IMBALANCE ESTIMATION USING UNBIASED TRAINING SEQUENCES,status allowed; assigned to the assignee hereof and hereby expresslyincorporated by reference herein.

The present application for patent is a continuation-in-part of patentapplication Ser. No. 11/684,566, filed Mar. 9, 2007, entitled,QUADRATURE MODULATION ROTATING TRAINING SEQUENCE, status pending;assigned to the assignee hereof and hereby expressly incorporated byreference herein.

The present application for patent is a continuation-in-part of patentapplication Ser. No. 11/755,719, filed May 30, 2007, entitled,QUADRATURE IMBALANCE MITIGATION USING UNBIASED TRAINING SEQUENCES,status pending, assigned to the assignee hereof and hereby expresslyincorporated by reference herein.

The present application for patent is related to U.S. patent applicationentitled, CHANNEL ESTIMATION USING FREQUENCY SMOOTHING, having Ser. No.11/853,809, filed concurrently herewith, and assigned to the assignee,which is hereby incorporated by reference in its entirety.

BACKGROUND

1. Field

This invention relates generally to communication channel estimationand, more particularly, to systems and methods for improving the use ofquadrature modulation unbiased training sequences in the training ofreceiver channel estimates, by removing quadrature imbalance errors.

2. Background

FIG. 1 is a schematic block diagram of a conventional receiver front end(prior art). A conventional wireless communications receiver includes anantenna that converts a radiated signal into a conducted signal. Aftersome initial filtering, the conducted signal is amplified. Given asufficient power level, the carrier frequency of the signal may beconverted by mixing the signal (down-converting) with a local oscillatorsignal. Since the received signal is quadrature modulated, the signal isdemodulated through separate I and Q paths before being combined. Afterfrequency conversion, the analog signal may be converted to a digitalsignal, using an analog-to-digital converter (ADC), for basebandprocessing. The processing may include a fast Fourier transform (FFT).

There are a number of errors that can be introduced into the receiverthat detrimentally affect channel estimations and the recovery of theintended signal. Errors can be introduced from the mixers, filters, andpassive components, such as capacitors. The errors are exacerbated ifthey cause imbalance between the I and Q paths. In an effort to estimatethe channel and, thus, zero-out some of these errors, communicationsystems may use a message format that includes a training sequence,which may be a repeated or predetermined data symbol. Using anOrthogonal Frequency Division Multiplexing (OFDM) system for example,the same IQ constellation point may be transmitted repeatedly for eachsubcarrier.

In an effort to save power in portable battery-operated devices, someOFDM systems use only a single modulation symbol for training. Forexample, a unique direction in the constellation (e.g., the I path) isstimulated, while the other direction (e.g., the Q path) is not. Thesame type of unidirectional training may also be used with pilot tones.Note: scrambling a single modulation channel (e.g., the I channel) with±1 symbol values does not rotate the constellation point, and providesno stimulation for the quadrature channel.

In the presence of quadrature path imbalance, which is prevalent inlarge bandwidth systems, the above-mentioned power-saving trainingsequence results in a biased channel estimate. A biased channel estimatemay align the IQ constellation well in one direction (i.e., the I path),but provide quadrature imbalance in the orthogonal direction. It ispreferable that any imbalance be equally distributed among the twochannels.

FIG. 2 is a schematic diagram illustrating quadrature imbalance at thereceiver side (prior art). Although not shown, transmitter sideimbalance is analogous. Suppose that the Q path is the reference. Theimpinging waveform is cos(ωt+θ), where θ is the phase of the channel.The Q path is down-converted with −sin(ωt). The I path is down-convertedwith (1+2ε)cos(ωt+2Δφ). 2Δφ and 2ε are hardware imbalances, respectivelya phase error and an amplitude error. The low pass filters H_(I) andH_(Q) are different for each path. The filters introduce additionalamplitude and phase distortion. However, these additional distortionsare lumped inside 2Δφ and 2ε. Note: these two filters are real andaffect both +ω and −ω in an identical manner.

Assuming the errors are small:(1+2ε)cos(ωt+2Δφ)≈(1+2ε)cos(ωt)−2Δφ· sin(ωt)

-   -   The first component on the right hand side, cos(ωt), is the        ideal I path slightly scaled. The second component, −2Δφ·        sin(ωt), is a small leakage from the Q path. After        down-conversion of the impinging waveform:    -   in the I path: (1+2ε)cos(θ)+2ε· sin(θ).    -   in the Q path: sin(θ).

The errors result in the misinterpretation of symbol positions in thequadrature modulation constellation, which in turn, results inincorrectly demodulated data.

SUMMARY

Wireless communication receivers are prone to errors caused by a lack oftolerance in the hardware components associated with mixers, amplifiers,and filters. In quadrature demodulators, these errors can also lead toimbalance between the I and Q paths, resulting in improperly processeddata.

A training signal can be used to calibrate receiver channels. However, atraining signal that does not stimulate both the I and Q paths, does notaddress the issue of imbalance between the two paths. An unbiasedtraining sequence can be used to stimulate both the I and Q paths, whichresults in a better channel estimate. Conventionally, channel estimatesare derived from predetermined information associated with the positive(+f) subcarriers. Even better channel estimates can be obtained if thenegative (−f) subcarriers are used to derive an estimate of any residualquadrature imbalance.

Accordingly, a method is provided for removing quadrature imbalanceerrors in received data. The method accepts an unbiased trainingsequence in a quadrature demodulation receiver. An unbiased trainingsequence has a uniform accumulated power evenly distributed in a complexplane, and includes predetermined reference signals (p) at frequency +fand predetermined mirror signals (p_(m)) at frequency −f. The unbiasedtraining sequence is processed, generating a sequence of processedsymbols (y) at frequency +f, representing complex plane information inthe unbiased training sequence. Each processed symbol (y) is multipliedby the mirror signal (p_(m)), and an unbiased quadrature imbalanceestimate B_(m) is obtained at frequency (−f).

For example, the unbiased training sequence may be accepted on a firstsubcarrier, and the quadrature imbalance estimate obtained for the firstsubcarrier. Then, the method accepts quadrature modulated communicationdata on the first subcarrier in symbol periods subsequent to acceptingthe unbiased training sequence. A processed symbol (y_(c)) is generatedfor each communication data symbol, and each processed symbol (y_(c)) ismultiplied by a quadrature imbalance estimate to derive animbalance-corrected symbol.

The method also multiplies the processed symbol (y) by a conjugate ofthe reference signal (p*) to obtain an unbiased channel estimate (h_(u))at frequency +f. Using the quadrature imbalance and channel estimates,imbalance-corrected symbols can be derived.

Additional details of the above-described method, and a system forremoving quadrature imbalance errors in received data are presentedbelow.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic block diagram of a conventional receiver front end(prior art).

FIG. 2 is a schematic diagram illustrating quadrature imbalance at thereceiver side (prior art).

FIG. 3 is a schematic block diagram depicting an exemplary datatransmission system.

FIG. 4 is a schematic block diagram of a system or device fortransmitting an unbiased communications training sequence.

FIG. 5A is a diagram depicting an unbiased training sequence representedin both the time and frequency domains.

FIGS. 5B and 5C are diagrams depicting the uniform accumulation of powerevenly distributed in a complex plane.

FIG. 6 is a diagram depicting an unbiased training sequence enabled as asequence of pilot tones in the time domain.

FIG. 7 is a diagram depicting an unbiased training sequence enabled as apreamble preceding non-predetermined communication data.

FIG. 8 is a diagram depicting an unbiased training sequence enabled byaveraging symbols over a plurality of messages.

FIG. 9 is a schematic block diagram of a system for removing quadratureimbalance errors in received data.

FIG. 10 depicts the performance achieved by applying the above-describedalgorithms to the WiMedia UWB standard.

FIGS. 11A and 11B are flowcharts illustrating a method for removingquadrature imbalance errors in received data.

DETAILED DESCRIPTION

Various embodiments are now described with reference to the drawings. Inthe following description, for purposes of explanation, numerousspecific details are set forth in order to provide a thoroughunderstanding of one or more aspects. It may be evident, however, thatsuch embodiment(s) may be practiced without these specific details. Inother instances, well-known structures and devices are shown in blockdiagram form in order to facilitate describing these embodiments.

As used in this application, the terms “processor”, “processing device”,“component,” “module,” “system,” and the like are intended to refer to acomputer-related entity, either hardware, firmware, a combination ofhardware and software, software, or software in execution. For example,a component may be, but is not limited to being, a process running on aprocessor, generation, a processor, an object, an executable, a threadof execution, a program, and/or a computer. By way of illustration, bothan application running on a computing device and the computing devicecan be a component. One or more components can reside within a processand/or thread of execution and a component may be localized on onecomputer and/or distributed between two or more computers. In addition,these components can execute from various computer readable media havingvarious data structures stored thereon. The components may communicateby way of local and/or remote processes such as in accordance with asignal having one or more data packets (e.g., data from one componentinteracting with another component in a local system, distributedsystem, and/or across a network such as the Internet with other systemsby way of the signal).

Various embodiments will be presented in terms of systems that mayinclude a number of components, modules, and the like. It is to beunderstood and appreciated that the various systems may includeadditional components, modules, etc. and/or may not include all of thecomponents, modules etc. discussed in connection with the figures. Acombination of these approaches may also be used.

The various illustrative logical blocks, modules, and circuits that havebeen described may be implemented or performed with a general purposeprocessor, a digital signal processor (DSP), an application specificintegrated circuit (ASIC), a field programmable gate array (FPGA) orother programmable logic device, discrete gate or transistor logic,discrete hardware components, or any combination thereof designed toperform the functions described herein. A general-purpose processor maybe a microprocessor, but in the alternative, the processor may be anyconventional processor, controller, microcontroller, or state machine. Aprocessor may also be implemented as a combination of computing devices,e.g., a combination of a DSP and a microprocessor, a plurality ofmicroprocessors, one or more microprocessors in conjunction with a DSPcore, or any other such configuration.

The methods or algorithms described in connection with the embodimentsdisclosed herein may be embodied directly in hardware, in a softwaremodule executed by a processor, or in a combination of the two. Asoftware module may reside in RAM memory, flash memory, ROM memory,EPROM memory, EEPROM memory, registers, hard disk, a removable disk, aCD-ROM, or any other form of storage medium known in the art. A storagemedium may be coupled to the processor such that the processor can readinformation from, and write information to, the storage medium. In thealternative, the storage medium may be integral to the processor. Theprocessor and the storage medium may reside in an ASIC. The ASIC mayreside in the node, or elsewhere. In the alternative, the processor andthe storage medium may reside as discrete components in the node, orelsewhere in an access network.

FIG. 3 is a schematic block diagram depicting an exemplary datatransmission system 300. A baseband processor 302 has an input on line304 to accept digital information form the Media Access Control (MAC)level. In one aspect, the baseband processor 302 includes an encoder 306having an input on line 304 to accept digital (MAC) information and anoutput on line 308 to supply encoded digital information in thefrequency domain. An interleaver 310 may be used to interleave theencoded digital information, supplying interleaved information in thefrequency domain on line 312. The interleaver 310 is a device thatconverts the single high speed input signal into a plurality of parallellower rate streams, where each lower rate stream is associated with aparticular subcarrier. An inverse fast Fourier transform (IFFT) 314accepts information in the frequency domain, performs an IFFT operationon the input information, and supplies a digital time domain signal online 316. A digital-to-analog converter 318 converts the digital signalon line 316 to an analog baseband signal on line 320. As described inmore detail below, a transmitter 322 modulates the baseband signal, andsupplies a modulated carrier signal as an output on line 324. Note:alternate circuitry configurations capable of performing the samefunctions as described above would be known by those with skill in theart. Although not explicitly shown, a receiver system would be composedof a similar set of components for reverse processing informationaccepted from a transmitter.

FIG. 4 is a schematic block diagram of a system or device fortransmitting an unbiased communications training sequence. The system400 comprises a transmitter or transmission means 402 having an input online 404 to accept digital information. For example, the information maybe supplied from the MAC level. The transmitter 402 has an output online 406 to supply a quadrature modulation unbiased training sequencerepresenting a uniform accumulated a power evenly distributed in acomplex plane.

The transmitter 402 may include a transmitter subsystem 407, such as aradio frequency (RF) transmitter subsystem that uses an antenna 408 tocommunicate via an air or vacuum media. However, it should be understoodthat the invention is applicable to any communication medium (e.g.,wireless, wired, optical) capable of carrying quadrature modulatedinformation. The transmitter subsystem 407 includes an in-phase (I)modulation path 410, or a means for generating I modulation traininginformation in the time domain having an accumulated power. Thetransmitter subsystem 407 also includes a quadrature (Q) modulation path412, or a means for generating Q modulation training information in thetime domain having an accumulated power equal to the I modulation pathpower. I path information on line 404 a is upconverted at mixer 414 withcarrier fc, while Q path information on line 404 b is upconverted atmixer 416 with a phase shifted version of the carrier (fc+90°). The Ipath 410 and Q path 412 are summed at combiner 418 and supplied on line420. In some aspects, the signal is amplified at amplifier 422 andsupplied to antenna 408 on line 406, where the unbiased trainingsequences are radiated. The I and Q paths may alternately be referred toas I and Q channels. A unbiased training sequence may also be referredto as a rotating training signal, a quadrature balanced trainingsequence, balanced training sequence, balanced training sequence, orunbiased training signal.

For example, the unbiased training sequence may be initially sent viathe I modulation path 410, with training information subsequently sentvia the Q modulation path 412. That is, the training signal may includeinformation, such as a symbol or a repeated series of symbols sent onlyvia the I modulation path, followed by the transmission of a symbol orrepeated series of symbols sent only via the Q modulation path.Alternately, training information may be sent initially via the Qmodulation path, and subsequently via the I modulation path. In the caseof single symbols being sent alternately through the I and Q paths, thetransmitter sends a rotating training signal. For example, the firstsymbol may always be (1,0), the second symbol may always be (0,1), thethird symbol (−1,0), and the fourth symbol (0,−1).

However, it is not necessary to simply alternate the transmission ofsymbols through the I and Q modulations paths to obtain symbol rotation,as described above. For example, the transmitter may send traininginformation simultaneously through both the I and Q modulation paths,and combine I and Q modulated signals.

The above-mentioned rotating type of unbiased training sequence, whichinitially sends training signal via (just) the I modulation path, may beaccomplished by energizing the I modulation path, but not energizing theQ modulation path. Then, the transmitter sends a training signal via theQ modulation path by energizing the Q modulation path, subsequent tosending training information via the I modulation path. The trainingsymbols can also be rotated by supplying symbols, each with both I and Qcomponents, as is conventionally associated with quadrature modulation.

Typically, the transmitter 402 also sends quadrature modulated(non-predetermined) communication data. The unbiased training sequenceis used by a receiver (not shown) to create unbiased channel estimates,which permit the non-predetermined communication data to be recoveredmore accurately. In one aspect, the quadrature modulated communicationdata is sent subsequent to sending the unbiased training sequence. Inanother aspect, the unbiased training sequence is sent concurrently withthe communication data in the form of pilot signals. The system is notlimited to any particular temporal relationship between the trainingsignal and the quadrature modulated communication data.

To be unbiased, the symbol values associated with any particularsubcarrier may periodically vary. The simplest means of evenlydistributing information in the complex plane when there are an evennumber of symbols per message, is to rotate the symbol value 90 degreesevery period. As used herein, a message is a grouping of symbols in apredetermined format. A message has a duration of several symbolsperiods. One or more symbols may be transmitted every symbol period.Some messages include a preamble preceding the main body of the message.For example, a message may be formed as a long packet containing manyOFDM symbols. Each OFDM symbol contains many subcarriers. In someaspects, the message preamble includes the unbiased training sequence.In other aspects, the unbiased training sequence is a sequence of pilotsignals that are transmitted simultaneously with the non-predeterminedcommunication data.

If an uneven number of symbols are used in the training sequence of amessage, a methodology that rotates the phase of the symbol by 90degrees every period is not always useful. For a sequence of 3 symbols,a 60-degree or 120-degree rotation may be used to evenly distribute thesymbol values in the complex plane. For 5 symbols, a 180/5-degree or360/5-degree rotation may be used. If the number of symbols in atraining sequence is a prime number, combination solutions can be used.For example, if there are a total of 7 symbols in a message, then arotation of 90 degrees may be used for the first 4 symbols, and arotation of 120 (or 60) degrees for the next three symbols. In anotheraspect, the unbiased training sequence may be averaged over more thanone message. For example, if a message includes 3 training symbols, thenthe combination of 2 messages includes 6 symbols. In the context of a6-symbol training signal, a rotation of 90 degrees may be used betweensymbols.

Since power is a measurement responsive to the squaring of a complexsymbol value, the power associated with a symbol vector at angle θ incomplex space may also be considered to be the power at (θ+180). Hence,the accumulated power at an angle of 60 degrees is the same as the powerat 240 degrees. Alternately stated, the power associated with a symbolat angle θ may be summed with the power at angle (θ+180). By summing thepower at angles θ and (θ+180), complex space, as considered from theperspective of power, only spans 180 degrees. For this reason, a uniformaccumulation of power is evenly distributed in complex space when theunbiased training sequence consists of only 2 orthogonal symbols, or 3symbols separated by 60 degrees.

FIG. 5A is a diagram depicting an unbiased training sequence representedin both the time and frequency domains. In one aspect the transmittergenerates a signal pair including a complex value reference signal (p)at frequency +f and a complex value mirror signal (p_(m)) at frequency−f, with a nullified product (p·p_(m)). For example, at time i=1, theproduct (p₁·p_(1m))=0. As noted above, p and p_(m) are complex valueswith amplitude and phase components. In another aspect, the transmittergenerates i occurrences of the reference signal (p) and mirror signal(p_(m)), and nullifies the sum of the products (p_(i)·p_(im)).Alternately stated, the sum of (p_(i)·p_(im))=0, for i=1 to N. Note: the“dot” between the p_(i) and p_(im) symbols is intended to represent aconventional multiplication operation between scalar numbers.

Likewise, when the transmitter generates i occurrences of the referencesignal and mirror signal, the signal pair values p and pm may, but neednot, vary for every occurrence. For example, the transmitter may nullifythe sum of the products (p_(i)·p_(im)) by generating information as acomplex value that remains constant for every occurrence, to representp. To represent p_(m), the transmitter may generate information as acomplex value that rotates 180 degrees every occurrence. However, thereare almost an infinite number of other ways that the products(p_(i)·p_(im)) may be nulled.

In another aspect, the transmitter generates i occurrences of referencesignal (p) and mirror signal (p_(m)), and a product (p_(i)·p_(im)) foreach occurrence. The transmitter pairs occurrences and nullifies the sumof the products from each paired occurrence.

For example, one or more messages may contain a temporal sequence of Npilot tones, for a given subcarrier f, with N pilot tones for the mirrorsubcarrier −f. As noted above in the discussion of FIG. 5A, to create anunbiased training sequence using this pilot tone, the general solutionis the sum of (p_(i)·p_(im))=0, for i=1 to N. For one particularsolution, the pilot tones are paired for i=1 and 2. Thus,p₁·p_(1m)+p₂·p_(2m)=0. Likewise, the pilot tones for i=3 and 4 may bepaired as follows: p₃·p_(3m)+p₄·p_(4m)=0. This pairing may be continuedout to i=N. If each pair has a sum of zero, then the total sum is alsozero, i.e., sum p_(i)·p_(im)=0. Pairing simplifies the nulling issue.Instead of searching for N pilots that verify sum p_(i)·p_(im)=0, it isenough that 2 pair of pilots can be nulled.

As described above, simple examples of creating an unbiased trainingsequence include either the rotation of symbols by 90 degrees in thetime domain, or in the frequency domain, maintaining the symbolreference on +f, but flipping the sign the mirror on −f. Both theseexamples used 2 pair of tones and satisfy the equationp₁·p_(1m)+p₂·p_(2m)=0.

Alternately expressed, the unbiased training sequence may include:

-   -   Time 1: p₁ for +f and p_(1m) for −f;    -   Time 2: p₂ for +f and p_(2m) for −f;    -   Time 3: p₃ for +f and p_(3m) for −f; and,    -   Time 4: p₄ for +f and p_(4m) for −f.

The unbiased training sequence can be obtained by averaging. Theprinciple of unbiased training sequence dictates that the pilot mustsatisfy:p ₁ ·p _(1m) +p ₂ ·p _(2m) +p ₃ ·p _(3m) +p ₄ ·p _(4m)=0

As a variation, the unbiased training sequence can be organized asfollows:p ₁ ·p _(1m) +p ₂ ·p _(2m)=0 and p ₃ ·p _(3m) +p ₄ ·p _(4m)=0

FIGS. 5B and 5C are diagrams depicting the uniform accumulation of powerevenly distributed in a complex plane. The complex plane can be used torepresent real axis (R) and imaginary axis (I) information. The circlerepresents the boundary of uniform power or energy with a normalizedvalue of 1. In FIG. 5B, the unbiased training sequence is formed from 3symbols: a first symbol (A) at 0 degrees; a second symbol (B) at 120degrees; and a third symbol (C) at 240 degrees. The exact same powerdistribution is obtained when the first symbol (A) remains at 0 degrees,the second symbol (B′) is at 60 degrees, and the third symbol (C′) is at120 degrees. The power associated with each symbol is 1.

In FIG. 5C, the unbiased training sequence is formed from 5 symbols: 2symbols at 0 degrees, each with a power of 0.5, so that the accumulatedpower is 1; a symbol at 90 degrees with a power of 1: a symbol at 180degrees with a power of 1; and a symbol at 270 degrees with a power of1.

As used herein, the above-mentioned “uniform accumulation of power” maybe exactly equal accumulations in each complex plane direction, as inmany circumstances it is possible to transmit and receive an unbiasedtraining sequence with an error of zero. That is, the training sequenceis 100% biased. Alternately stated, the sum of p_(i)·p_(im)=0, asdescribed above. In a worst case analysis, L pilot symbols are averaged,each having a uniform accumulated power as follows:|sum p _(i) ·p _(im)|=sum|pi| ² =L

-   -   If L is 100%, and if a |sum p_(i)·p_(im)|=L/4, then the (uniform        accumulated power) error is 25%. An unbiased training sequence        with a 25% error still yields excellent results. If L/2 is used        (a 50% error), good results are obtained as the IQ interference        from the channel estimate still decreases by 6 dB.

FIG. 6 is a diagram depicting an unbiased training sequence enabled as asequence of pilot tones in the time domain. The transmitter may generatethe unbiased training sequence by supplying P pilot symbols per symbolperiod, in a plurality of symbol periods. Each pulse in the figurerepresents a symbol. The transmitter generates (N−P) quadraturemodulated communication data symbols per symbol period, andsimultaneously supplies N symbols per symbol period, in the plurality ofsymbol periods. Many communications systems, such as those compliantwith IEEE 802.11 and UWB use pilot tones for channel training purposes.

FIG. 7 is a diagram depicting an unbiased training sequence enabled as apreamble preceding non-predetermined communication data. The transmittergenerates quadrature modulated communication data and supplies theunbiased training sequence in a first plurality of symbol periods (e.g.,at times 1-4), followed by the quadrature modulated communication datain a second plurality of symbol periods (e.g., at times 5 through N).Again, the pulses in the figure represent symbols.

For example, an Ultra Wideband (UWB) system uses 6 symbols transmittedprior to the transmission of communication data or a beacon signal.Therefore, 3 consecutive symbols may be generated on the I modulationpath followed by 3 consecutive on the Q modulation path. Using thisprocess, the Q channel need only be activated briefly, for 3 symbols,before returning to sleep. However, there are many other combinations ofsymbols that may be used to generate an unbiased training sequence.

Viewing either FIG. 5B or 5C, it can be seen that the transmittergenerates a temporal sequence of complex plane symbols with equalaccumulated power in a plurality of directions (in the complex plane).As used herein, “direction” refers to the summation of vectors at eachangle θ and (θ+180). For example, the power associated with a symbol at0 degrees is accumulated with the power from a symbol at 180 degrees, as0 and 180 degrees are the same direction. As a consequence of thisrelationship, the temporal sequence of symbols in the unbiased trainingsequence have a cumulative power associated with real axis informationin the time domain, and an equal cumulative power associated withimaginary axis information in the time domain, as supplied in aplurality of symbols periods by the transmitter. In another aspect, theunbiased training sequence representing the uniform accumulated powerevenly distributed in the complex plane may be expressed as a temporalsequence of i complex symbols (a) in the time domain, as follows:sum a _(i)(k)·a _(i)(k)=0

-   -   where k is a number of samples per symbol period. Note: the        “dot” between the a_(i) and a_(i) symbols is intended to        represent a conventional multiplication operation between scalar        numbers.

Since the symbol a_(i) is typically a subcarrier with a periodicwaveform, there is no one particular value for a. That is, a_(i) varieswith time, and could be represented as a_(i)(t). However, if t samplesare obtained, the symbol may be expressed as a_(i)(kT), or a_(i)(k),assuming T is normalized to 1. For time domain systems, the summationover k disappears. With only one sample per symbol, the symbol andsample become the same and the equation can be written as:sum a _(i) ·a _(i)=0

To illustrate with a simple 2-symbol orthogonal unbiased trainingsequence, if the first symbol (i=1) has an angle of 0 degree, an equalamount of power must exist at an angle of 180 degrees in order tosatisfy the equation. Likewise, if the second symbol is at 90 degrees,and equal amount of power must exist at an angle of 270 degrees. Othermore complication examples may require that the symbols be summed overthe index of i to obtain the nulled final result.

Alternately considered, the formula sum a_(i)·a_(i)=0 refers to the factthat if a projection is made in any direction in the complex plane andthe power calculated, the power is always the same, regardless of theangle. The power in direction φ is:sum|Re a _(i)(−jφ)|²=0.5 sum|a _(i)|²+0.5 Re(−2jφsum a _(i) a _(i)=0

-   -   This power is constant for all φ if and only if sum        a_(i)·a_(i)=0.

It can be shown that the frequency domain formula (sum p_(i)·p_(im)=0)is equivalent to sum a_(i)·a_(i)=0. The time domain signal correspondingto p_(i) and p_(im) is:a _(i) =p _(i)exp(j2πft)+p _(im)exp(−2πft)

-   -   since p_(i) modulates +f and p_(im) modulates −f.    -   Within one symbol i, the integral over time of a_(i)·a_(i) is:        integral a _(i) ·a _(i)=integral {pi·piexp(j4πft)+p _(im) ·p        _(im)exp(−j4πft)+p _(i) ·p _(im) }=p _(i) ·p _(im)    -   since the exp(j4πft) rotates several times and vanishes when        integrated in one symbol.    -   So a_(i)·a_(i) cumulated in one symbol is equal to p_(i)·p_(im).    -   If all the symbols are added up:        sum integral a _(i) ·a _(i)=sum p _(i) ·p _(im)=0

FIG. 8 is a diagram depicting an unbiased training sequence enabled byaveraging symbols over a plurality of messages. A symbol (or more thanone, not shown) is generated in a first symbol period in a firstmessage. A symbol is generated in a second symbol period in a secondmessage, subsequent to the first message. More generally, a traininginformation symbols are generated in a plurality (n) messages. Thetransmitter generates the unbiased training sequence by creating equalpower in a plurality of complex plane directions, as accumulated overthe plurality of messages. Although a preamble type training sequence isshown, similar to FIG. 7, the same type of analysis can be applied topilot-type unbiased training sequence.

FIG. 9 is a schematic block diagram of a system for removing quadratureimbalance errors in received data. The system or device 900 comprises aquadrature demodulation receiver or receiving means 902 having an inputon line 904 to accept an unbiased training sequence. As with thetransmitter of FIG. 4, the receiver 902 may be an RF device connected toan antenna 905 to receive radiated information. However, the receivermay alternately receive the unbiased training sequence via a wired oroptical medium (not shown).

The receiver 902 has an in-phase (I) demodulation path 906 for acceptingI demodulation training information in the time domain having anaccumulated power. A quadrature (Q) demodulation path 908 accepts Qdemodulation training information in the time domain. When consideringthe unbiased training sequence, the Q path has an accumulated powerequal to the I modulation path power. As is conventional, the receiver902 includes analog- to digital converters (ADC) 909, a fast Fouriertransformer (FFT) 910, a deinterleaver 912, and a decoder 914.

The quadrature demodulation receiver 902 accepts an unbiased trainingsequence of predetermined reference signals (p) at frequency (+f) andpredetermined mirror signals (p_(m)) at frequency (−f) with a uniformaccumulated power evenly distributed in a complex plane. The receiver902 generates a sequence of processed symbols (y) at frequency (+f)representing complex plane information in the unbiased trainingsequence, multiplies each processed symbols (y) by the mirror signal(pm), and supplies a quadrature imbalance estimate (B_(m)) at frequency(−f). For simplicity, it is shown that the generation of the processedsymbols and the multiplication by the mirror symbols is performed insymbol generator 916.

As noted above in the discussion of the transmitter, the unbiasedtraining sequence is a temporal sequence of complex plane symbols withequal accumulated power in a plurality of directions. Alternatelyconsidered, the receiver accepts the unbiased training sequence as asignal pair including a complex value reference signal (p) at frequency+f and a complex value mirror signal (p_(m)) at frequency −f, where theproduct (p·p_(m)) is null. For example, the receiver accepts theunbiased training sequence as i occurrences of the reference signal (p)and the mirror signal (p_(m)), where the sum of the products(p_(i)·p_(im)) is null.

In some aspects, the receiver 902 accepts an unbiased training sequencewith a plurality of simultaneously accepted predetermined referencesignals (p_(n)) and a plurality of simultaneously accepted predeterminedmirror signals (p_(nm)). For example, n pilot symbols may be acceptedevery symbol period. The receiver 902 generates a plurality of processedsymbols (y_(n)) from the corresponding plurality of reference signals,multiplies each processed symbol by its corresponding mirror signal, andobtains a plurality of channel estimates (B_(nm)) from the correspondingplurality of (y_(n))(p_(nm)) products.

More explicitly (see FIG. 6), the receiver accepts the unbiased trainingsequence as P pilot symbols per symbol period, in a plurality of symbolperiods, and obtains P unbiased pilot channel estimates. Simultaneously,the receiver accepts (N−P) quadrature modulated communication datasymbols in each symbol period and generates a processed symbol (y_(c))for communication data in each symbol period (depicted as Y_(N−P)).Channels estimates are extrapolated for each processed symbol (y_(c)),and quadrature imbalance estimates B_(m) (depicted as (B_(m))_(N−P)) arederived for each processed symbol (y_(c)) from the pilot channelquadrature imbalance estimates (depicted as (B_(m))_(1-P)).

In another aspect, the receiver accepts an unbiased training sequencewith temporal sequence of n predetermined reference signals (p_(n)) andn predetermined mirror signals (p_(nm)), see FIG. 5A. The receivergenerates a temporal sequence of n processed symbols (y_(n)) from thetemporal sequence of reference signals, and multiplies each processedsymbol in the temporal sequence by its corresponding mirror signal. Atemporal sequence of n quadrature imbalance estimates (B_(nm)) isobtained and the n quadrature imbalance estimates are averaged.

More explicitly as shown in FIG. 7, the receiver may accept the unbiasedtraining sequence on a first subcarrier and the receiver derives aquadrature imbalances estimate (B_(m)) for the first subcarrier. Thereceiver accepts quadrature modulated communication data on the firstsubcarrier in symbol periods subsequent to accepting the unbiasedtraining sequence, generating a processed symbol (y_(c)) for eachcommunication data symbol. A quadrature imbalance estimate (B_(m)) isderived from each processed symbol.

Returning to FIG. 9, the receiver (i.e., symbol generator 916)multiplies the processed symbol (y) by a conjugate of the referencesignal (p*), obtains an unbiased channel estimate (h_(u)) at frequency+f. Further, the unbiased training sequence is processed to generate asequence of processed symbols (y_(m)) at frequency −f. The receivermultiplies symbol (y_(m)) by (p_(m)*) to obtain channel estimate h_(m),at frequency −f, and multiplies symbol y_(m) by p* to obtain quadratureimbalance estimate B at frequency +f.

The receiver calculates an imbalance-corrected symbol(z)=y−(B_(m)/h_(m)*)y_(m)*, if the signal-to-noise ratio (SNR) of(x_(m)) is greater than j, and otherwise sets (z) equal to (y). Forsimplicity, zero-forcing (ZF) calculator 918 is shown supplying theimbalance-corrected symbols in response to receiving processed symbols,channel estimates, and quadrature imbalance estimates. The receiver(i.e., ZF calculator 918) calculates (z_(m))=y_(m)−(B/h*)y*, if the SNRof (x) is greater than j, and otherwise, sets (z_(m)) equal to (y_(m)).The receiver uses (z) and (z_(m)) in the calculation of (x) and (x_(m)),respectively, which is outside the scope of this disclosure. In oneaspect, as explained in greater detail below, the receiver calculates(z_(m)) and (z) using the quadrature imbalance estimates (B) and(B_(m)), respectively, if the SNR is greater than 1 (j=1).

Although not specifically shown, the receiver of FIG. 9 may also beenabled as a processing device for removing quadrature imbalance errorsin received data. Such a processing device comprises a quadraturedemodulation receiving module having an input to accept an unbiasedtraining sequence of predetermined reference signals (p) at frequency(+f) and predetermined mirror signals (p_(m)) at frequency (−f) with auniform accumulated power evenly distributed in a complex plane. Thereceiving module generates a sequence of processed symbols (y) atfrequency (+f) representing complex plane information in the unbiasedtraining sequence, multiplies each processed symbols (y) by the mirrorsignal (pm), and supplies a quadrature imbalance estimate (B_(m)) atfrequency (−f).

Training sequences, whether enabled in a preamble or as pilot signalsare similar in that the information content of transmitted data istypically predetermined or “known” data that permits the receiver tocalibrate and make channel measurements. When receiving communication(non-predetermined) data, there are 3 unknowns: the data itself, thechannel, and noise. The receiver is unable to calibrate for noise, sincenoise changes randomly. Channel is a measurement commonly associatedwith delay and multipath. For relatively short periods of time, theerrors resulting from multipath can be measured if predetermined data isused, such as training or pilot signals. Once the channel is known, thismeasurement can be used to remove errors in received communication(non-predetermined) data. Therefore, some systems supply a trainingsignal to measure a channel before data decoding begins.

However, the channel can change, for example, as either the transmitteror receiver moves in space, or the clocks drift. Hence, many systemscontinue to send more “known” data along with the “unknown” data inorder to track the slow changes in the channel.

Although not specifically shown, the transmitter of FIG. 4 and thereceiver of FIG. 9 may be combined to form a transceiver. In fact, thetransmitter and receiver of such a transceiver may share elements suchas an antenna, baseband processor, and MAC level circuitry. Theexplanations made above are intended to describe a transceiver that bothtransmits unbiased training sequences and calculates unbiased channelestimates based upon the receipt of unbiased training sequences fromother transceivers in a network of devices.

Functional Description

Modern high data rate communication systems transmit signals on twodistinct channels, the in-phase and quadrature-phase channels (I and Q).The two channels form a 2D constellation in a complex plane. QPSK andQAM are examples of constellations. The I and Q channels may be carriedby RF hardware that cannot be perfectly balanced due to variations in RFcomponents, which results in IQ imbalance. In the increasingly commondirect conversion systems, the imbalance issued are even greater. IQimbalance distorts the constellation and results in crosstalk betweenthe I and Q channels: the signal interferes with itself. Increasingtransmission power does not help, since self-generated interferenceincreases with the signal power. The signal-to-noise ratio (SINR)reaches an upper bound that puts a limit on the highest data rateattainable with a given RF hardware. In order to increase the data rate,a costly solution is to use fancier, more expensive hardware. A possiblyless costly solution is to digitally estimate IQ imbalance andcompensate for it. The concepts of digital estimation and compensationalgorithms have been previously advanced in the art. However, thesolutions tend to be expensive because they do not rely on a specialtype of training sequence. These solutions often only consider imbalanceat one side, usually at the receiver.

Examples are given below that focus on Orthogonal Frequency DivisionMultiplexing (OFDM), with insights for time domain systems, which studyend-to-end imbalance, from transmitter to receiver. Moreover, in OFDMthe imbalance is modeled as a function of frequency, taking into accountvariations in the frequency response of the filters.

Two kinds of enhancements are presented: one with zero cost thateliminates the interference from the channel estimate by using anunbiased training sequence. Substantial gains are achieved because theerror of the channel estimate is often more detrimental to performancethan the error in the data itself. A second, relatively low cost,enhancement compensates for data distortion, if more gain is needed.

A model of the IQ imbalance is provided below. Analysis is provided toshow how conventional channel estimation using unbiased trainingsequences can mitigate part of the IQ imbalance. Then, a straightforwardextension is provided to calculate the IQ imbalance parameters, provingthat the algorithms are effective. Using the estimated parameters, asimple compensation algorithm is presented to mitigate data distortion.Simulation results for WiMedia's UWB are also given, as well assuggestions to amend the standard.

IQ Imbalance Model

IQ imbalance arises when the power (amplitude) balance or theorthogonality (phase) between the in-phase (I) and quadrature-phase (Q)channels is not maintained. IQ imbalance is therefore characterized byan amplitude imbalance 2ε and a phase imbalance 2Δφ.

Time Domain Signals

A complex symbol x is transmitted and received via the I and Q channels.In an ideal noiseless channel, the symbol x is received intact. But inthe presence of IQ imbalance, a noisy or distorted version is likelyreceived.y=αx+βx*  (1)whereα=cos(Δφ)+jε sin(Δφ)β=ε cos(Δφ)−j sin(Δφ)  (2)are complex quantities modeling the imbalance, α≈1 and β≈0. Nonlinearmodel (1) is linearized via the vector form

$\begin{matrix}{\begin{pmatrix}y \\y^{*}\end{pmatrix} = {{{\begin{pmatrix}a \\\beta^{*}\end{pmatrix}\begin{pmatrix}\chi \\\chi^{*}\end{pmatrix}}->Y} = {{BX}.}}} & (3)\end{matrix}$B is the imbalance matrix. The second row is obsolete since it is aduplicate version of the first row. But it gives a same size and typeinput and output so imbalance blocks at transmitter and receiver can beconcatenated, as described below. The imbalance matrix at thetransmitter is defined by B_(t), and at the receiver it is defined byB_(r).

One-Tap Channel

A one-tap channel is considered, suitable for OFDM. A one-tap channel hin appropriate matrix form is

$\begin{matrix}{H = \begin{pmatrix}h & 0 \\0 & h^{*}\end{pmatrix}} & (4)\end{matrix}$With imbalance at transmitter and receiver, and in average whileGaussian (AWGN) noise n, vector form N=(ηη*)T, the received signal isexpressed as a concatenation of linear blocks

$\begin{matrix}{{\begin{matrix}{Y = {{B_{r}{HB}_{t}X} + N}} \\{\overset{\Delta}{=}{{H^{\prime}X} + N}} \\{\overset{\Delta}{=}{{\begin{pmatrix}h^{\prime} & \beta^{\prime} \\\beta^{\prime*} & h^{\prime*}\end{pmatrix}\begin{pmatrix}x \\x^{\prime}\end{pmatrix}} + \begin{pmatrix}n \\n^{*}\end{pmatrix}}}\end{matrix}->y} = {{h^{\prime}x} + {\beta^{\prime}x^{*}} + {n.}}} & (5)\end{matrix}$The overall result is that IQ imbalance and channel combine to create aglobal channel h′, plus an undesired distortion or interferencecharacterized by a global imbalance parameter β′. The global imbalanceparameter β′ changes when the channel changes, and may need to beestimated regularly.

Next, the condition is considered where the symbol x, rather thanspanning the entire complex plane, is restricted to a given (1D) axis.For example, the axis may be associated with BPSK modulation, the realaxis, the imaginary axis, or any axis in between. In this case, x*=kxmay be written, where k is a complex constant (a rotation), and

$\begin{matrix}\begin{matrix}{y = {{( {h^{\prime} + {\beta^{\prime}k}} )x} + n}} \\{\overset{\Delta}{=}{{h^{''}x} + {n.}}}\end{matrix} & (6)\end{matrix}$If x is restricted to a unique axis, IQ imbalance vanishes, becoming anintegral part of an overall channel response.

Frequency Domain Signals

While the previous model applies to time domain signals, a modificationis now considered where the signal of interest x is given in frequencydomain, at frequency f. In time domain, this signal is carried by acomplex tone, xe^(j2πft). Replacing terms in equation (1), the followingis obtainedαxe ^(j2πft) +βx*e ^(−j2πft)  (7)In OFDM, the interference created by IQ imbalance does not show up atthe same frequency f, but rather at the mirror frequency −f, and viceversa. What is transmitted at −f creates interference on frequency +f.If signal x_(m) is the signal transmitted at frequency −f, where index mdenotes a quantity at mirror frequency −f, then at frequency −f thefollowing is obtainedα_(m) x _(m) e ^(−j2πft)+β_(m) x _(m) *e ^(j2πft)  (8)A generalization of the time domain equations has been used. The IQimbalance parameters α and β are here a function of frequency. Thismodels an imbalance due to different low-pass (base-band) or band-pass(IF) filters in the system. The I and Q paths cannot have the exact samefilters and, hence, the imbalance varies with frequency. In time domainsystems, this kind of imbalance exists but it is very expensive tocompensate. An equalizer and an extension of the model to deal withdifferent convolutions on different channels are required. So in thetime domain, bulk or average imbalance is used. Frequency domain systemsare able to take advantage of the plain equalizer structure and modelthe imbalance on a per frequency basis.

If the output of equations (7) and (8) are combined per subcarrier, thefollowing is observedY=(αx+β _(m) x _(m)*)e ^(j2πft)y _(m)=(α_(m) x _(m) +βx*)e ^(−j2πft)  (9)Omitting the subcarriers (automatically handled by the FFT), a linearmodel function of signals at +f and −f can be written as

$\begin{matrix}{{\underset{\_}{\begin{pmatrix}y \\y_{m}^{*}\end{pmatrix} = {\begin{pmatrix}a & \beta_{m} \\\beta^{*} & a_{m}^{*}\end{pmatrix}\begin{pmatrix}x \\x_{m}^{*}\end{pmatrix}}}->Y} = {BX}} & (10)\end{matrix}$In the frequency domain model, the second row is no longer obsolete. Themodel deals, in one shot, with a pair of mirror frequencies. A one-tapchannel h at frequency f, and h_(m) at frequency −f is modeled by thematrix

$\begin{matrix}{H = {\begin{pmatrix}h & 0 \\0 & h_{m}^{*}\end{pmatrix}.}} & (11)\end{matrix}$AWGN noise n at frequency f, and n_(m) at frequency −f form the noisevector N=(ηη′_(m))^(T). The end to end model is

$\begin{matrix}{\begin{matrix}{Y = {{B_{r}{HB}_{t}X} + N}} \\{\overset{\Delta}{=}{{H^{\prime}X} + N}}\end{matrix}\mspace{20mu}{{\underset{\_}{\overset{\Delta}{=}{{\begin{pmatrix}h^{\prime} & \beta_{m}^{\prime} \\\beta^{\prime*} & h^{\prime*}\end{pmatrix}\begin{pmatrix}x \\x_{m}^{*}\end{pmatrix}} + \begin{pmatrix}n \\n_{m}^{*}\end{pmatrix}}}->y} = {{h^{\prime}x} + {\beta_{m}^{\prime}x_{m}^{*}} + n}}{y_{m} = {{h_{m}^{\prime}x_{m}} + {\beta^{\prime}x^{*}} + n_{m}}}} & (12)\end{matrix}$h′, h_(m)′ are the global channel taps, and β′, β_(m)′ are the globalimbalance parameters. The imbalance parameters change when the channelschange and may need to be estimated regularly.

Since IQ imbalance generates interference exclusively from the mirrorfrequency, two interesting cases are noteworthy. If at the mirrorfrequency no signal is transmitted, or the channel is in a fade, nointerference is created. If on the other hand, the signal or channel isstrong, the interference can be strong. Hence, in OFDM, the effect of IQimbalance is more problematic.

Conventional Channel Estimation

Before examining the compensation algorithms, it is shown how half ofthe problem can be solved at no cost, simply by using an unbiasedtraining sequence. An unbiased training sequence fully eliminates theinterference from the channel estimate, noticeably improvingperformance. In fact, the error in the channel estimate is often moredetrimental than the error in the data, because the channel estimatetends to create a bias in the constellation.

The model (12) is stimulated with pilot tones. At frequency +f, thepilot p is transmitted, and at frequency −f, the pilot p_(m). Assuming,without loss of generality, that the pilots have a unit norm (thechannel carries the effective power), the conventional channel estimateat frequency f is obtained by de-rotating by p*

$\begin{matrix}\begin{matrix}{\hat{h} = {{h^{\prime}{pp}^{*}} + {\beta_{m}^{\prime}p_{m}^{*}p^{*}} + n}} \\{= {h^{\prime} + {\beta_{m}^{\prime}p_{m}^{*}p^{*}} + n}}\end{matrix} & (13)\end{matrix}$By averaging several channel observations, the noise is automaticallyreduced (for clarity, noise de-rotation is omitted). With regard to theterm β′_(m)p_(m)*p*, many OFDM systems (e.g., WiMedia's UWB) use atraining sequence that is simply a repeated symbol. Therefore, this termdoes not decay with averaging. Applying a scrambling of +1 or −1 to theentire OFDM symbol does not help, as nothing changes when the sign ofboth p* and p_(m)* are inverted. Rather, the following is accomplished:after cumulating a number of observations, the sum of the products isnullifiedΣ_(i) p _(i) p _(im)=0  (14)Often the training sequence consists of an even number of symbols, andit is enough to ensure each pair adds up to zerop ₁ p _(1m) +p ₂ p _(2m)=0  (15)

TABLE 1 Examples of unbiased training sequences P₂ = jp₁ Second trainingsymbol is a 90 degrees rotation of first training symbol. P₂ = p₁,p_(2m) = −p_(1m) For positive frequencies maintain fixed pilot, fornegative frequencies constantly invert the sign.

Examples of simple sequences that satisfy the condition are given inTable 1. These types of training sequences are denoted as unbiasedtraining sequences because, on one hand, unbiased channel estimates areproduced, and on the other, the training signals equally spans the I andQ dimensions of the complex plane in time domain. For example, anunbiased training sequence is not concentrated along just the real axis.

As a proof: consider the unit norm complex scalara_(i)=p_(i)e^(jθ)=p_(im)e^(−jθ), half way between p_(i) and p_(im). Intime domain, the pilots add up to 2a_(i) cos(2πft+θ). In time domain andin a given OFDM symbol, the 2 mirror pilots span a unique directiondetermined by the complex constant a_(i). If L symbols are transmitted,the total (or average, or cumulated) power in a direction φ is Σ_(i)|

a_(i) exp(−jφ)|²=0.5 L+0.5

exp(−2 jφ) Σa_(i)a_(i). This power is constant in any direction φ if andonly if Σ_(i)a_(i)a_(i)≡Σ_(i)p_(i)p_(im)=0. Uniform spanning of thecomplex plane is achieved.

IQ Imbalance Estimation

After estimating the global channel h′, the estimation of the globalimbalance parameter β_(m)′ is considered. Careful analysis of equation(12) reveals that this parameter can be obtained in manner very similarway to the conventional channel estimation. That is, β_(m)′ can betreated like a “channel” carrying the pilot p_(m)*. Hence, byde-rotating by p_(m), an estimate of the imbalance may be obtained. Thecondition for unbiased estimation of the imbalance is identical toequation (14).

In summary, using unbiased training sequences and two conventionalchannel estimations, good estimates of the end-to-end channel andimbalance parameter are obtained (Table 2).

TABLE 2 Estimation algorithm H′ β′_(m) Derotate by p* Derotate by p_(m)

Smoothing Over Adjacent Subcarriers

In addition to averaging over adjacent OFDM symbols, the channelestimate may be smoothed over adjacent subcarriers within one symbol. InOFDM, the cyclic prefix is designed to be short, and the channel issupposed to vary slowly from tone to tone. Likewise, the filters in theRF chain should have short temporal response and their frequencyresponse also varies slowly, i.e., the IQ imbalance varies slowly acrosssubcarriers. The same channel smoothing techniques can be used to smoothand improve the imbalance parameter estimate. By using unbiased trainingsequences, there is no interaction between the channel estimate and theimbalance estimate. Each estimated can be independently smoothed.

If a unique OFDM symbol is used for estimation, it is impossible to findan unbiased training sequence that satisfies equation (14). In thiscase, a nearly unbiased training sequence can be obtained by applyingthe summation from equation (14) over groups of 2 or more adjacentsubcarriers. Then smoothing automatically cancels all or part of theinterference from mirror frequencies. One solution is to rotate thepilot by 90 degrees on the adjacent subcarrier (moving in mirrordirections on the positive and negative frequencies).

Estimation

The use of unbiased training sequences and the above-mentionedconventional channel estimation results is a Least Squares (LS)estimator. Of all the LS estimators, the Minimum Mean Squared Error(MMSE) sense shows significant value.

Least Squares Estimator

L transmissions X_(i), L noise terms N_(i) and L observations Y_(i), maybe respectively concatenated into the 2 by L matricesχ=(X ₁ X ₂ . . . X _(L))N=(N ₁ N ₂ . . . N _(L))γ=(Y ₁ Y ₂ . . . Y _(L))  (16)Then, equation (12) becomesγ=H′ _(χ) +N  (17)The unknown is H′. The LS estimator isĤ′=YX ^(H)(XX ^(H))⁻¹  (18)When condition (14) is satisfied, it is easy to verify that χχ^(H) isdiagonal (the cross terms vanish). It is proportional to an identitymatrix since the pilots are normalized to unit norm. ThenĤ′=YX ^(H) /L=1/LΣi Y ₁ X ₁ ^(H)  (19)is precisely four conventional channel estimations with de-rotationsrespectively by p_(i)*, p_(im), p_(im)* and p_(i) as described in theprevious section. Two estimations are obtained for frequency f, and twoestimations for mirror frequency −f.

Optimal Estimator

Unbiased training sequences and conventional channel estimations are anLS estimator. But any estimator Ĥ′=YX^(H)(XX^(H))⁻¹ is also an LSestimator. Below, it is shown that the use of unbiased trainingsequences results in an excellent estimator. Model (17) can be viewed asunknown information H′ sent via 2 consecutive transmissions over 2vectors (rows of χ) in an L dimension space. We denote by X_(f), N_(f),and Y_(f) respectively row j of} X, N and Y, where j ε {1,2}. Models(12) and (17) can be writteny ₁ =h′χ ₁+β′_(m)χ₂ +N ₁y ₂=β′χ₁ +h′ _(m)χ₂ +N ₂  (20)

There are 2 transmissions, each involving the 2 vectors X₁, X₂, andwhere each vector is carrying complex amplitude information to beestimated. The LS estimator consists of projecting onto each vector, ina parallel way to the other vector in order to cancel interference. Avery good result is obtained when the 2 vectors are orthogonal, i.e.,when dot product (14) is zero. Unbiased training sequences are bydefinition, training sequences that verify this condition. Othersequences use non-orthogonal vectors and suffer a loss of performancefunction of the angle between the vectors X₁ and X₂. Many OFDM systemscurrently use a very poor kind of training sequences where X₁, X₂ arecollinear, and it is impossible to properly estimate the 4 entries inH′. These training sequences tend to estimate noisier versions of thechannels h′ and h′_(m).

To calculate the Mean Squared Errors (MSE), the estimation error isĤ′−H′=NX^(H)(XX^(H))⁻¹. This is a 2 by 2 matrix, i.e., 4 error values.Each value can be isolated by multiplying left and right withcombinations of the vectors (1 0)^(T) and (1 0)^(T). Assuming ENN^(H) isan identity matrix, or more generally a diagonal matrix with elements σ²and σ_(m) ², it can be shown that the MSE of ĥ′ and {circumflex over(β)}′_(m) are, respectively, the first and second diagonal elements ofσ²(χχ^(H))⁻¹. And for β′ and ĥ_(m)′, the MSE are, respectively, thefirst and second diagonal element of σ_(m) ²(χχ^(H))⁻¹.

The total MSE is 2(σ²+σ_(m) ²)tr(χχ^(H))⁻¹. Now the problem is to find χthat minimizes tr(χχ^(H))⁻¹ subject to the constraint that total pilotpower is constant, i.e., tr(χχ^(H))=2 L. Using an Eigen decomposition,the problem can be written as minimize Σ1/λ_(j) subject to Σλ_(j) isconstant. The problem is solved with the Lagrange multipliers, and istypically optimum when all Eigen values are equal. This means χχ^(H)=LIis proportional to an identity matrix.

The total MSE has been minimized, and the resulting MSE per element iseither σ²/L or σ_(m) ²/L. But this MSE per element is likely to be thebest that can be obtained, even if a unique vector transmission is used.The MSE is unlikely to be improved for a 2 vector transmissions, andtherefore the MSE per element has been minimized. The unbiased trainingsequences plus conventional channel estimator are the MMSE of all LSestimators.

IQ Imbalance Compensation

If the gain from the unbiased channel estimate is not enough, the IQimbalance parameters may be estimated (as described previously) andapplied to compensate for data distortion. H′ is estimated in model(12), Y=H′X+N. Now the focus turns to the unknown data X. The model isthe same as any 2-tap channel with cross-correlations. Any channelequalization algorithm can be fitted. A simple equalization algorithm ispresented suitable for the ubiquitous bit-interleaved coded QAM andfading channels.

One concern with the Zero-Forcing (ZF) approach H′⁻¹Y=X+H′⁻¹N is that itenhances noise when the mirror channel is weak, unless an accounting ismade for the complicated colored noise. The present solution uses ZF,but only when the mirror channel is not weak. In equation (12),replacing x_(m) by its value, the following is obtainedy=(h′−β _(m) ′β′*/h _(m)′*)x+(β_(m) ′/h _(m)′*)y _(m)*−(β_(m) ′/h_(m)′*)n _(m) *+n≈h′x+(β_(m) ′/h _(m)′*)y _(m) *+n′+n  (21)where n′

−(β_(m)′/h_(m)′*)n_(m)* is noise enhancement. Note: it is assumed thesecond order imbalance term β′*β_(m)′<<h′h_(m)′*. When thisapproximation is invalid, the corrected channel h′_(c)

h′−β_(m)′β′*/h_(m)′* is considered, which entails precise estimation ofthe channel and imbalance parameters.

Basically, the ZF technique consists of computingz=y−(β_(m) ′/h _(m)′*)y _(m) *≈h′x+n′+n  (22)By subtracting the mirror frequency quantity (β_(m)′/h_(m)′)y_(m) fromthe received signal y, the simple channel model with no IQ imbalance isobtained. The rest of the decoding chain is unchanged.

This solution works well as long as the noise enhancement is weaker thanthe original interference from IQ imbalance, i.e.,|n′|²<|β_(m)′x_(m)*|². If not, then the original y is used rather thanthe imbalance corrected z. It is unnecessary to estimate n′ in order tomake a decision. A robust average-wise improvement may be elected. So,considering the expected values

$\begin{matrix}{{E{n^{\prime}}2} = {{{( {{{\beta\; m^{\prime}}}{2/{{h\; m^{\prime}}}}2} )E{{n\; m}}2} < {{{\beta\; m^{\prime}}}2E{{x\; m^{*}}}2}}->{{{{h\; m^{\prime}}}2^{\frac{E{x_{m}^{*}}2}{E{\eta_{m}}2}}\frac{E{x_{m}^{*}}2}{E{\eta_{m}}2}{\bullet{SNRm}}} > 1.}}} & (23)\end{matrix}$When the mirror frequency's signal to noise ratio SNR_(m) is greaterthan 1, the imbalance corrected term z is used. Otherwise, the originalsignal y is kept. Due to channel and imbalance estimation imprecision,it is safer to use a larger SNR, for example, SNR_(m)>2 works well forWiMedia UWB. Note that SNR_(m) can usually be obtained from the globalSNR via the formula SNR_(m)=|h_(m)′|²SNR.

Table 3 summarizes the ZF algorithm with noise enhancement avoidance.

TABLE 3 Compensation algorithm SNR_(m) < 1 + δ SNR_(m) > 1 + δ z = y z =y − (β_(m)′/h_(m)′)y_(m)Simulation Results

FIG. 10 depicts the performance achieved by applying the above-describedalgorithms to the WiMedia UWB standard. The highest data rate, 480 Mbps,is simulated in IEEE 802.15.3's channel model CM2 (indoorpico-environment of about 4 meters). Shadowing and band hopping areturned off. The IQ imbalance is constant and equal to 2ε=10% (0.8 dB) inamplitude and 2Δφ=10 degrees in phase. The same amount of imbalance ispresent at the transmitter and receiver. The figure shows the PacketError Rate (PER) as a function of Eb/No. The performance degradesquickly without any form of compensation. Table 4 lists the loss ofvarious algorithms with respect to ideal case.

TABLE 4 WiMedia UWB: loss from IQ imbalance at PER of 10⁻² CurrentStandard Unbiased Training Compensation 3.1 dB 1.1 dB 0.35 dB

End-to-end IQ imbalance and channel combine to form a global 2 by 2channel matrix. The use of unbiased training sequences achievesconsiderable gains at no cost. The unbiased training sequencesautomatically cancel end-to-end self-generated interference from thechannel estimate. Moreover, such training sequences are ideal forestimating IQ imbalance parameters, and a simple algorithm is given tocompensate for data distortion: Zero-Forcing with noise enhancementavoidance.

WiMedia UWB, in particular, benefits from the following enhancement: theconventional biased training sequence that consists of 6 symbolsexclusively transmitted on the I channel can be divided in 2 halves tocreate an unbiased sequence. The first 3 symbols are sent on the Ichannel, and the last 3 symbols are sent on the Q channel. By uniformlyspanning the complex plane, an unbiased training sequence is createdwith large gains for high data rates. For backward compatibility, thisscheme may be reserved for high data rate modes and signaled via thebeacons, or the training sequence type may be blindly detected.

In OFDMA (e.g., WiMAX), the subcarriers f and −f can be assigned todifferent users. Considerable interference can arise if power controldrives one user to high power level. It is therefore a good idea tolocate the pilots of different users on mirror subcarriers. The pilotsshould satisfy the unbiased training sequence criterion. Each userautomatically benefits without any extra effort. The pilots may hop todifferent locations while maintaining mirror positions.

The time domain formulas can be extended to Code Division MultipleAccess (CDMA) with a Rake equalizer combining several one-tap channels.Unbiased training sequences automatically improve the channel estimateper tap. A simple unbiased training sequence for CDMA consists ofconstantly rotating the complex symbols by 90 degrees.

FIGS. 11A and 11B are flowcharts illustrating a method for removingquadrature imbalance errors in received data. Although the method isdepicted as a sequence of numbered steps for clarity, the numbering doesnot necessarily dictate the order of the steps. It should be understoodthat some of these steps may be skipped, performed in parallel, orperformed without the requirement of maintaining a strict order ofsequence. As used herein, the terms “generating”, “deriving”, and“multiplying” refer to processes that may be enabled through the use ofmachine-readable software instructions, hardware, or a combination ofsoftware and hardware. The method starts at Step 1100.

Step 1102 accepts an unbiased training sequence in a quadraturedemodulation receiver. The unbiased training sequence has a uniformaccumulated power evenly distributed in a complex plane, and includespredetermined reference signals (p) at frequency +f and predeterminedmirror signals (p_(m)) at frequency −f. As explained in detail above,the unbiased training sequence is a temporal sequence of complex planesymbols with equal accumulated power in a plurality of directions.Alternately, Step 1102 accepts a signal pair including a complex valuereference signal (p) at frequency +f and a complex value mirror signal(p_(m)) at frequency −f, where the product (p·p_(m)) is null. Forexample, i occurrences of the reference signal (p) and the mirror signal(p_(m)) may be accepted, where the sum of the products (p_(i)·p_(im)) isnull.

Step 1104 processes the unbiased training sequence, generating asequence of processed symbols (y) at frequency +f, representing complexplane information in the unbiased training sequence. Step 1106multiplies each processed symbol (y) by the mirror signal (p_(m)). Step1108 obtains an unbiased quadrature imbalance estimate B_(m) atfrequency −f.

In one aspect, Step 1102 accepts an unbiased training sequence with aplurality of simultaneously accepted predetermined reference signals anda plurality of simultaneously accepted predetermined mirror signals(p_(nm)). Likewise, Step 1104 generates a plurality of signals (y_(n))from the corresponding plurality of reference signals (p_(n)). Step 1106multiplies each received symbol (y_(n)) by its corresponding mirrorsignal (p_(nm)), and Step 1108 obtains a plurality of unbiasedquadrature imbalance estimates (B_(nm)) from the corresponding pluralityof (y_(n))(p_(nm)) products.

For example, Step 1102 may accept P pilot symbols per symbol period, ina plurality of symbol periods, and Step 1108 obtains P pilot channelquadrature imbalance estimates per symbol period. In this aspect, Step1103 simultaneously accepts (N−P) quadrature modulated communicationdata symbols in each symbol period (also see FIG. 6). Then, generatingprocessed symbols in Step 1104 includes generating a processed symbol(y_(c)) for communication data in each symbol period. Likewise, derivingquadrature imbalance estimates in Step 1108 includes deriving quadratureimbalance estimates (B_(m)) for each processed symbol (y_(c)) from thepilot channel quadrature imbalance estimates.

In another aspect, Step 1102 accepts a temporal sequence of npredetermined mirror signals (p_(nm)) and n predetermined referencesignals (p_(n)). Generating the sequence of processed symbols (y) inStep 1104 includes generating a temporal sequence of n processed symbols(y_(n)). Then, obtaining the unbiased quadrature imbalance estimate(B_(nm)) in Step 1108 includes obtaining a sequence of n quadratureimbalance estimates, and averaging the n quadrature imbalance estimates.

For example (see FIG. 7), Step 1102 may accept the unbiased trainingsequence on a first subcarrier, and Step 1108 obtains the quadratureimbalance estimate for the first subcarrier. Then, Step 1110 acceptsquadrature modulated communication data on the first subcarrier insymbol periods subsequent to accepting the unbiased training sequence.Step 1112 generates a processed symbol (y_(c)) for each communicationdata symbol, and Step 1114 derives a quadrature imbalance estimates(B_(m)) for each processed symbol (y_(c)).

In other aspect the method includes the following additional steps. Step1116 multiplies the processed symbol (y) by a conjugate of the referencesignal (p*). Step 1118 obtains an unbiased channel estimate (h) atfrequency +f. Processing the unbiased training sequence in Step 1104includes generating a sequence of processed symbols (y_(m)) at frequency−f. Then, Step 1120 multiplies symbol (y_(m)) by (p_(m)*) to obtainchannel estimate h_(m), at frequency (−f), and Step 1122 multipliessymbol y_(m) by p* to obtain quadrature imbalance estimate B atfrequency +f.

If the signal-to-noise ratio (SNR) of (x_(m)) is greater than j (Step1124), then Step 1126 calculates an imbalance-corrected symbol(z)=y−(B_(m)/h_(m)*)y_(m)*. Otherwise, Step 1128 sets (z) equal to (y).If the SNR of (x) is greater than j (Step 1130), then Step 1132calculates (z_(m))=y_(m)−(B/h*)y*. Otherwise, Step 1134 sets (z_(m))equal to (y_(m)). Step 1136 uses (z) and (z_(m)) in the calculation of(x) and (x_(m)), respectively. In one aspect, j=1.

The above-described flowchart may also be interpreted as an expressionof a machine-readable medium having stored thereon instructions forremoving quadrature imbalance errors in received data. The instructionswould correspond to Steps 1100 through 1136, as explained above.

Systems, methods, devices, and processors have been presented to enablethe removal of quadrature imbalance errors in received data. Examples ofparticular communications protocols and formats have been given toillustrate the invention. However, the invention is not limited tomerely these examples. Other variations and embodiments of the inventionwill occur to those skilled in the art.

What is claimed is:
 1. A method of communication, the method comprising:receiving a training signal at a quadrature demodulation receiver, thetraining signal representing a plurality of predetermined referencevalues at a frequency +f or a frequency adjacent to the frequency +f anda plurality of corresponding predetermined mirror values at a frequency−f or a frequency adjacent to the frequency −f, wherein the sum of theproducts of each predetermined reference value and the correspondingpredetermined mirror value is zero; generating a plurality of receivedvalues based on the received portion of the training signal representingthe predetermined reference values at the frequency +f or the frequencyadjacent to the frequency +f; generating a plurality of derotationvalues by multiplying each received value by the correspondingpredetermined mirror value; and averaging the derotation values so as toobtain a quadrature imbalance estimate for the frequency −f or thefrequency adjacent to the frequency −f.
 2. The method of claim 1,wherein the received training signal represents at least a firstpredetermined reference value at the frequency +f, a secondpredetermined reference value at the frequency adjacent to the frequency+f, a first corresponding predetermined mirror value at the frequency−f, and a second corresponding predetermined mirror value at thefrequency adjacent to the frequency −f and wherein the sum of the firstpredetermined reference value multiplied by the first correspondingpredetermined mirror value and the second predetermined reference valuemultiplied by the second corresponding predetermined mirror value iszero.
 3. The method of claim 1, wherein the received training signalrepresents at least a first predetermined reference value at thefrequency +f received at a first time, a second predetermined referencevalue at the frequency +f received at a second time after the firsttime, a first corresponding predetermined mirror value at the frequency−f received at the first time, and a second corresponding predeterminedmirror value at the frequency −f received at the second time and whereinthe sum of the first predetermined reference value multiplied by thefirst corresponding predetermined mirror value and the secondpredetermined reference value multiplied by the second correspondingpredetermined mirror value is zero.
 4. The method of claim 1, furthercomprising: generating a plurality of second derotation values bymultiplying each received value by the conjugate of the correspondingpredetermined reference value; and averaging the second derotationvalues so as to obtain a channel estimate for the frequency +f or thefrequency adjacent to the frequency +f.
 5. The method of claim 1,further comprising: generating a plurality of second received valuesbased on the received portion of the training signal representing thepredetermined mirror values at the frequency −f or the frequencyadjacent to the frequency −f; generating a plurality of third derotationvalues by multiplying each second received value by the correspondingpredetermined reference value; and averaging the third derotation valuesso as to obtain a quadrature imbalance estimate for the frequency +f orthe frequency adjacent to the frequency +f.
 6. The method of claim 1,further comprising: generating a plurality of second received valuesbased on the received portion of the training signal representing thepredetermined mirror values at the frequency −f or the frequencyadjacent to the frequency −f; generating a plurality of fourthderotation values by multiplying each second received value by theconjugate of the predetermined mirror value; and averaging the fourthderotation values so as to obtain a channel estimate for the frequency−f or the frequency adjacent to the frequency −f.
 7. The method of claim1, further comprising: receiving a data signal at the quadraturedemodulation receiver, the data signal representing at least a pluralityof data values at frequency +f; generating a plurality of received datavalues based on the received portion of the data signal representing thedata values at the frequency +f; and generating a plurality ofcompensated data values based on the received data values and thequadrature imbalance estimate.
 8. The method of claim 7, wherein:receiving a data signal comprises receiving data signal representing aplurality of data values at frequency +f and a plurality ofcorresponding mirror data values at frequency −f; generating a pluralityof received data values further comprises generating a plurality ofreceived corresponding mirror data values based on the received portionof the data signal representing the corresponding mirror data values atfrequency −f; and generating a plurality of compensated data valuescomprises subtracting, from each received data value, a valueproportional to the quadrature imbalance estimate and the receivedcorresponding mirror data value.
 9. The method of claim 7, furthercomprising determining that signal-to-noise ratio is greater than athreshold, wherein generating a plurality of compensated data valuescomprises generating a plurality of compensated data values in responseto the determination.
 10. A system for communication, the systemcomprising: a receiver configured to receive a training signal, thetraining signal representing a plurality of predetermined referencevalues at a frequency +f or a frequency adjacent to the frequency +f anda plurality of corresponding predetermined mirror values at a frequency−f or a frequency adjacent to the frequency −f, wherein the sum of theproducts of each predetermined reference value and the correspondingpredetermined mirror value is zero; and a processor configured to:generate a plurality of received values based on the received portion ofthe training signal representing the predetermined reference values atthe frequency +f or the frequency adjacent to the frequency +f, generatea plurality of derotation values by multiplying each received value bythe corresponding predetermined mirror value, and average the derotationvalues so as to obtain a quadrature imbalance estimate for the frequency−f or the frequency adjacent to the frequency −f.
 11. The system ofclaim 10, wherein the received training signal represents at least afirst predetermined reference value at the frequency +f, a secondpredetermined reference value at the frequency adjacent to the frequency+f, a first corresponding predetermined mirror value at the frequency−f, and a second corresponding predetermined mirror value at thefrequency adjacent to the frequency −f and wherein the sum of the firstpredetermined reference value multiplied by the first correspondingpredetermined mirror value and the second predetermined reference valuemultiplied by the second corresponding predetermined mirror value iszero.
 12. The system of claim 10, wherein the received training signalrepresents at least a first predetermined reference value at thefrequency +f received at a first time, a second predetermined referencevalue at the frequency +f received at a second time after the firsttime, a first corresponding predetermined mirror value at the frequency−f received at the first time, and a second corresponding predeterminedmirror value at the frequency −f received at the second time and whereinthe sum of the first predetermined reference value multiplied by thefirst corresponding predetermined mirror value and the secondpredetermined reference value multiplied by the second correspondingpredetermined mirror value is zero.
 13. The system of claim 10, whereinthe processor is further configured to generate a plurality of secondderotation values by multiplying each received value by the conjugate ofthe corresponding predetermined reference value and average the secondderotation values so as to obtain a channel estimate for the frequency+f or the frequency adjacent to the frequency +f.
 14. The system ofclaim 10, wherein the processor is further configured to: generate aplurality of second received values based on the received portion of thetraining signal representing the predetermined mirror values at thefrequency −f or the frequency adjacent to the frequency −f, generate aplurality of third derotation values by multiplying each second receivedvalue by the corresponding predetermined reference value, and averagethe third derotation values so as to obtain a quadrature imbalanceestimate for the frequency +f or the frequency adjacent to the frequency+f.
 15. The system of claim 10, wherein the processor is furtherconfigured to: generate a plurality of second received values based onthe received portion of the training signal representing thepredetermined mirror values at the frequency −f or the frequencyadjacent to the frequency −f, generate a plurality of fourth derotationvalues by multiplying each second received value by the conjugate of thepredetermined mirror value, and average the fourth derotation values soas to obtain a channel estimate for the frequency −f or the frequencyadjacent to the frequency −f.
 16. The system of claim 10, wherein thereceiver is further configured to receive a data signal representing atleast a plurality of data values at frequency +f and the processor isfurther configured to generate a plurality of received data values basedon the received portion of the data signal representing the data valuesat the frequency +f and generate a plurality of compensated data valuesbased on the received data values and the quadrature imbalance estimate.17. The system of claim 16, wherein the data signal represents aplurality of data values at frequency +f and a plurality ofcorresponding mirror data values at frequency −f and the processor isconfigured to generate a plurality of received corresponding mirror datavalues based on the received portion of the data signal representing thecorresponding mirror data values at frequency −f and to generate theplurality of compensated data values by subtracting, from each receiveddata value, a value proportional to the quadrature imbalance estimateand the received corresponding mirror data value.
 18. The system ofclaim 16, wherein the processor is configure to determine thatsignal-to-noise ratio is greater than a threshold and to generate theplurality of compensated data values in response to the determination.19. A system for communication, the system comprising: means forreceiving a training signal at a quadrature demodulation receiver, thetraining signal representing a plurality of predetermined referencevalues at a frequency +f or a frequency adjacent to the frequency +f anda plurality of corresponding predetermined mirror values at a frequency−f or a frequency adjacent to the frequency −f, wherein the sum of theproducts of each predetermined reference value and the correspondingpredetermined mirror value is zero; means for generating a plurality ofreceived values based on the received portion of the training signalrepresenting the predetermined reference values at the frequency +f orthe frequency adjacent to the frequency +f; means for generating aplurality of derotation values by multiplying each received value by thecorresponding predetermined mirror value; and means for averaging thederotation values so as to obtain a quadrature imbalance estimate forthe frequency −f or the frequency adjacent to the frequency −f.
 20. Acomputer-readable non-transitory medium having instructions encodedthereon which, when executed by a computer, cause a system to perform amethod of communication, the method comprising: receiving a trainingsignal at a quadrature demodulation receiver, the training signalrepresenting a plurality of predetermined reference values at afrequency +f or a frequency adjacent to the frequency +f and a pluralityof corresponding predetermined mirror values at a frequency −f or afrequency adjacent to the frequency −f, wherein the sum of the productsof each predetermined reference value and the correspondingpredetermined mirror value is zero; generating a plurality of receivedvalues based on the received portion of the training signal representingthe predetermined reference values at the frequency +f or the frequencyadjacent to the frequency +f; generating a plurality of derotationvalues by multiplying each received value by the correspondingpredetermined mirror value; and averaging the derotation values so as toobtain a quadrature imbalance estimate for the frequency −f or thefrequency adjacent to the frequency −f.